| L(s) = 1 | − 6·3-s + 4·5-s − 4·7-s + 27·9-s + 28·11-s + 26·13-s − 24·15-s − 36·17-s + 44·19-s + 24·21-s + 8·23-s − 226·25-s − 108·27-s + 204·29-s + 164·31-s − 168·33-s − 16·35-s + 668·37-s − 156·39-s − 100·41-s − 272·43-s + 108·45-s − 60·47-s − 566·49-s + 216·51-s + 708·53-s + 112·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.357·5-s − 0.215·7-s + 9-s + 0.767·11-s + 0.554·13-s − 0.413·15-s − 0.513·17-s + 0.531·19-s + 0.249·21-s + 0.0725·23-s − 1.80·25-s − 0.769·27-s + 1.30·29-s + 0.950·31-s − 0.886·33-s − 0.0772·35-s + 2.96·37-s − 0.640·39-s − 0.380·41-s − 0.964·43-s + 0.357·45-s − 0.186·47-s − 1.65·49-s + 0.593·51-s + 1.83·53-s + 0.274·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.976405656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.976405656\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 242 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 582 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2426 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 2038 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 44 T - 498 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 204 T + 52270 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 164 T + 66294 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 668 T + 212814 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T - 2230 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 272 T + 137142 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 60 T + 203746 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 708 T + 312478 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 180 T + 395626 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1068 T + 726830 T^{2} - 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 420 T + 207854 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 436 T + 749474 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 412 T + 163398 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 672 T + 559646 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 124 T + 501530 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 140 T + 1088138 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 188 T - 343530 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.639134782559222820705833742138, −8.479360228838093779193916470292, −7.88335187427264595991189219480, −7.70914663507645320890912779024, −6.97813181518322587517449558663, −6.79590162055459088814552887895, −6.29379436037329359840412513949, −6.18098324769395376523947301197, −5.67712785077041955867717631772, −5.44435872815901686370455578666, −4.64263068239161157146201960069, −4.60547752360649103893513416751, −3.92425760569416123010242257170, −3.79960137742305090801049765714, −2.78466172469384241970553352331, −2.75160647357195239929332145033, −1.71088368964779093207535195851, −1.55136222901196661982519524271, −0.70686582964195287517465519461, −0.53627557505848555798188301609,
0.53627557505848555798188301609, 0.70686582964195287517465519461, 1.55136222901196661982519524271, 1.71088368964779093207535195851, 2.75160647357195239929332145033, 2.78466172469384241970553352331, 3.79960137742305090801049765714, 3.92425760569416123010242257170, 4.60547752360649103893513416751, 4.64263068239161157146201960069, 5.44435872815901686370455578666, 5.67712785077041955867717631772, 6.18098324769395376523947301197, 6.29379436037329359840412513949, 6.79590162055459088814552887895, 6.97813181518322587517449558663, 7.70914663507645320890912779024, 7.88335187427264595991189219480, 8.479360228838093779193916470292, 8.639134782559222820705833742138
